Consider the two payment plans as shown here for an $18,000 auto loan.
Plan 1: Monthly payment of $306.91 over a period of 5 years, which corresponds to a
compound interest rate of 0.9% compounded monthly.
Plan 2: Monthly payment of $229.02 over a period of 5 years, which corresponds to a
compound interest rate of 1.9% compounded monthly.
Determine which plan has a lower cost of credit. Find the lower credit cost.
A. Plan 2 has a lower cost of credit, which is $414.60.
B. Plan 2 has a lower cost of credit, which is $823.08.
C. Plan 1 has a lower cost of credit, which is $414.60.
D. Plan 1 has a lower cost of credit, which is $1,534.55
To find the lower cost of credit, we need to calculate the total amount paid for each payment plan and subtract the original loan amount.
For plan 1, the monthly payment is $306.91 and the loan term is 5 years, which corresponds to 5*12 = 60 months.
Using the compound interest formula:
A = P(1+r/n)^(nt)
P = loan amount = $18,000
r = interest rate per year = 0.009 (0.9%)
n = number of times interest is compounded per year = 12
t = number of years = 5
Using the formula, the total amount paid for plan 1 is:
A = 18000(1+0.009/12)^(12*5)
A = 18000(1+0.00075)^60
A ≈ 18000(1.00075)^60
A ≈ 18000(1.0463)
A ≈ $18,833.40
The cost of credit for plan 1 is the total amount paid minus the original loan amount:
Cost of credit = $18,833.40 - $18,000 = $833.40
For plan 2, the monthly payment is $229.02 and the loan term is 5 years, which corresponds to 5*12 = 60 months.
Using the same compound interest formula with the new interest rate:
A = P(1+r/n)^(nt)
P = loan amount = $18,000
r = interest rate per year = 0.019 (1.9%)
n = number of times interest is compounded per year = 12
t = number of years = 5
Using the formula, the total amount paid for plan 2 is:
A = 18000(1+0.019/12)^(12*5)
A = 18000(1+0.0015833)^60
A ≈ 18000(1.09325)
A ≈ $19,677.00
The cost of credit for plan 2 is the total amount paid minus the original loan amount:
Cost of credit = $19,677.00 - $18,000 = $1,677.00
Comparing the two costs of credit, we see that plan 1 has a lower cost of credit.
However, option D is incorrect as it incorrectly states that the lower cost of credit for plan 1 is $1,534.55.
Therefore, the correct answer is C. Plan 1 has a lower cost of credit, which is $414.60.